Сборник задач по высшей математике 2 том
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3.4.33. |
Bbl'mCJIHTbMacCY np5lMoyrOJIbHOrO napaJIJIeJIemme)J;a 0 ~ x ~ a, |
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o ~ y ~ b, 0 ~ z ~ c, eCJIH nJIOTHOCTb B TO'IKe(x, y, z) nponopU;H- |
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OHaJIbHa cyMMe Koop)J;HHaT 9TOil: TO'IKH. |
3.4.34. |
Onpe)J;eJIHTb MacCY mapa pa)];Hyca R, nJIOTHOCTb KOToporo nponop- |
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U;HOHaJIbHa paCCT05lHHIO OT u;eHTpa mapa, npH'IeMHa paCCT05lHHH |
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e)J;HHHU;bI OT u;eHTpa nJIOTHOCTb paBHa )J;ByM. |
3.4.35. |
Hail:TH Maccy TeJIa, OrpaHH'IeHHOrOnOBepXHOCT5IMH z = h H |
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x 2 + y2 = Z2, |
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eCJIH nJIOTHOCTb B KroK)J;Oil: TO'IKenponOpU;HOHaJIbHa annJIHKaTe |
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9TOil: TO'IKH. |
3.4.36. |
Hail:TH MacCY c<pepH'IeCKOrOCJI051 Me:>K)J;y c<pepaMH X2+y2+Z2 = a2 |
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H x 2 + y2 + Z2 = 4a2, eCJIH nJIOTHOCTb B KroK)J;Oil: TO'IKeo6paTHo |
nponOpU;HOHaJIbHa paCCT05lHHIO TO'IKHOT Ha'IaJIaKoop)J;HHaT.
B'bt"tuc.n,umb o6r>eM'bt me.n" 02pa'H.u"te'H.'H.'btx n06epX'H.OC'T11JlMU:
3.4.37.Z = x + y, Z = xy, x + y = 1, x = 0, y = O.
3.4.38.x 2 + Z2 = a2, x + y = ±a, x - y = ±a.
3.4.39. az = x 2 + y2, Z = ) x 2 + y2, a > O.
3.4.40.(x2 + y2 + Z2)2 = 2az, x 2 + y2 = Z2.
3.4.41. |
X = 0, y = 0, Z = 0, 2x - 3y - 12 = 0, 2z = y2. |
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3.4.42. |
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3.4.43. |
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3.4.44. |
z = 1; _ x 2 - |
y2, Z = 15)x2 + y2. |
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3.4.45. |
Z = )4 - x 2 - |
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y2, Z = |
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255. |
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3.4.46. |
z = )64 - |
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x 2 - |
y2, x 2 + y2 ~ 60, z = l. |
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3.4.47. |
x 2 + y2 = y, x 2 + y2 = 4y, z = )x2 + y2, z = O. |
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3.4.48. |
X2 + y2 |
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3.4.49. |
!dx |
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dz. |
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3.4.50. |
!dz |
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! (Z2 + y2) dx. |
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3.4.51. |
/dy |
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Vy2 + z 2 dx. |
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-a |
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y2 |
l!..(y2+Z2) |
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a2 |
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3.4.52. |
/ dx |
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.jZdz. |
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B'bt"I:uC.ltUmb mpoil:H'bte U'Hmezpa.lt'bt: |
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3.4.53. |
///ZVX2 +y2 dxdydz, r.n;e 06JIacTh V 3a.n;aHa |
HepaBeHCTBaMH |
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°:::;; X :::;; 2, °:::;; |
y :::;; |
V2x - |
X2, °:::;; Z :::;; a. |
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3.4.54. |
/ / / |
xyz2 dxdydz, r.n;e V JIe)KHT B I-M OKTaHTe H OrpaHH'feHae.n;H- |
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HH'fHOil:c<pepoil: x 2 + y2 + Z2 = 1 H Koop.n;HHaTHhIMH IIJIOCKOCTjiMH |
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X = 0, y = 0, Z = 0. |
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3.4.55. |
/ / / |
2y2eXy dxdydz, r.n;e V OrpaHH'feHaIIJIOCKOCTjiMH X = 0, y = 1, |
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z = 0, Z = l. |
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Y = x, |
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X = 1, y = 0, |
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3.4.56. |
/ / / |
X dxdydz, |
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V OrpaHH'feHaIIJIOCKOCTjiMH |
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YV= lOx, z =°H IIapa6oJIOH.n;OM Z = xy.
3.4.57./ / / x 2z sin(xyz) dxdydz, r.n;e V OrpaHH'feHaIIJIOCKOCTjiMH X = 0,
XV= 2, y = 0, y = 7r Z = 0, Z = l.
3.4.58. / / / 8y2 zexyz dxdydz, r.n;e V OrpaHH'feHaIIJIOCKOCTjiMH x - I,
V
X = 0, y = 0, y = 2, z = 0, Z = l.
3.4.59./ / /(x + y + z) dxdydz, r.n;e V 3a,ll;aHa HepaBeHCTBaMH °:::;; X :::;; a,
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°:::;; |
y :::;; b, °:::;; z :::;; |
c. |
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V 3a,ll;aHa HepaBeHCTBaMH °:::;; <p |
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3.4.60. |
///PSin()dpd<pd(), |
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:::;; ~, |
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P :::;; 2, °:::;; () :::;; |
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°:::;; |
~. |
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3.4.61. |
/ xdxdydz, r.n;e V OrpaHH'feHaIIJIOCKOCTjiMH X = 0, y = 0, z = 0, |
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Y = h, X + z = a. |
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OrpaHH'feHaIIJIOCKOCTjiMH |
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3.4.62. |
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z = 0, |
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(1 + !f + 1l. + |
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z = 8 (1 - ~ - ~), X = 0, y = 0.
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B-WtUCAUmb MaCC'bt OaHOpOaH'btX meA, 02paHU"teHH'btX n06epXHOC'ln.RMU:
3.4.63.x 2 + y2 + 4z2 = l.
3.4.64.x + y + z = a, x + y + z = 2a, x + y = z, x + y = 2z.
3.4.65. |
y2 = 4a2 - |
3ax, y2 = ax, z = ±h. |
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3.4.66. |
y2 |
z2 |
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+ ""2 = 2Ci , x = a. |
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Haitmu 'lCOOpaUHam'bt '4eHmpa 'In.R'JICecmu OaHOpOaH'btX meA, 02paHU"teHH'btX n06epXHOC'ln.RMU:
3.4.67. |
IIJIOCKOCTlIMH X |
= 0, y = 0, z = 0, 2x + 3y - 12 = 0 H U;HJIHH,ll;POM |
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y2 |
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z=2' |
= 0, x + z = 6 H U;HJIHH,ll;paMH z = ..;x H Z = 2..;x. |
3.4.68. |
IIJIOCKOCTlIMH z |
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3.4.69. |
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x2 +y2 |
C¢epoil: x 2 + y2 + Z2 = 3a2 H rrapa60JIOH,ll;OM Z = 2a (Ha,ll; |
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HHM). |
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3.4.70. |
C¢epoil: x 2 + y2 + z2 = R2 H KOHYCOM ztga = Jx2 + y2, tga > ° |
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(Ha,ll; KOHYCOM).
Haitmu MOMeHm'bt UHep'4UU OaHOpOaH'btX me.!! C aaHHoit Maccoit M:
3.4.71. IIpllMoyrOJIbHoro rrapaJIJIeJIerrHrre,ll;a C pe6paMH a, b, C OTHOCHTeJIbHO KroK,ll;Oro H3 pe6ep H OTHOCHTeJIbHO CBoero u;eHTpa TIDKeCTH. IIIapa pa,ll;Hyca R OTHOCHTeJIbHO rrpllMoil:, KaCaTeJIbHoil: K rnapy. OJIJIHrrCOH,ll;a
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~+1L+~=1
a2 b2 c2
OTHOCHTeJIbHO KroK,ll;Oil: H3 Tpex CBOHX oceil:.
Hail:TH CTaTHgeCKHe MOMeHTbI OTHOCHTeJIbHO KOOp,ll;HHaTHbIX rrJIOCKocTeil: H KOOp,ll;HHaTbI u;eHTpa TIDKeCTH O,ll;HOpO,ll;HOrO TeJIa, orpaHHgeHHoro rrapa60JIOH,ll;OM z = 3 - x 2 - y2 H rrJIOCKOCTbIO Z = 0.
KOHTponbHble BonpOCbl III 60nee CnO)l(Hbie 3aA3HIIISI
B'bt"tucAumb 06r>eM'bt meA, 02paHU"teHH'btX n06epXHOC'ln.RMU:
3.4.75. IIJIOCKOCTlIMH x=O, x+y=2, x-y=2 H U;HJIHH,ll;paMH z=ln(x+2)
H Z = In(6 - x).
3.4.76. IIJIOCKOCTbIO Z = X + Y H rrapa60JIOH,ll;OM Z = x 2 + y2. 3.4.77. IIJIOCKOCTbIO 2x + Z = 2 H rrapa60JIOH,ll;OM (x - 1)2 + y2 = Z.
3.4.78.(x 2 + y2 + Z2)2 = a2(x2 + y2 _ z2).
3.4.79.(x2 + y2 + Z2)3 = 3xy.
3.4.80.C¢epoil: x 2 + y2 + Z2 = 4 H rrapa60JIOH,ll;OM x 2 + y2 = 3z.
3.4.81.(x 2 + y2 + Z2)2 = axyz.
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3.4.82. f f f y2(exy - e- xy ) dxdydz, V orpruuPleHa IIOBepXHOCT5!MM x = 0,
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y = -2, y = 4x, z = 0, z = 2. |
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3.4.83. |
f f f(y2 + Z2) dxdydz, V OrpaHM'IeHaIIJIOCKOCT5!MM x = 0, y = 0, |
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zv= 0, x + y = 1, z = x + y. |
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3.4.84. |
f f f xy2 z3 dxdydz, V OrpaHM'IeHa |
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3.4.85. |
fff(X~ + y: + Z~) dxdydz, V: x~ + y: + z~ :::;; l. |
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3.4.86. |
' r!-=-_-;=;=dX;;:d=y=d=;;z=~::;;:, |
V: x2 + y2 + Z2 :::;; l. |
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/ f,- |
1 + J(x 2 + y2 + Z2)3 |
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3.4.87.(x 2 + y2 + z2)2 = a3x.
3.4.88.(x 2 + y2 + Z2)2 = a2z4.
3.4.89.(x2 + y2 + z2)3 = a2(x2 + y2)2.
3.4.90.(x 2 + y2)2 + Z4 = a3z.
Ha11mu MOMe'Hm'bt U'Hep-quu om'Hocume.!!'b'HO ?l;OOpaU'Ham'H'btx n.t/,oc?l;ocme11 oa-
'HOpOa'H'btX me.!!, ozpa'HU"I,e'H'H'btX n06epXHOC1l'l.R.MU: |
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3.4.91. |
x 2 |
y2 |
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°bOO) |
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+ b2 |
C' |
a + b = |
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> , > |
,c > . |
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3.4.92. |
2" |
+ b2 |
2" = 1, |
2" + b2 = |
a (a > 0). |
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3.4.93. |
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HaitTM MOMeHT MHepU;MM OTHOCMTeJIhHO OCM Oz O,ll;HOpO,ll;HOrO TeJIa, |
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OrpaHM'IeHHoroIIOBepXHOCT5!MM y = :2 x 2 , z = 0, |
z = %(b - y) |
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(a> 0, b > 0, h > 0). |
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3.4.94. |
HaitTM MOMeHT MHepU;MM OTHOCMTeJIhHO OCM Oz O,ll;HOpO,ll;HOrO TeJIa, |
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OrpaHMqeHHOrO IIOBepXHOCT5!MM z = ~ (y2 - |
x2), Z = 0, y = ±a. |
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Ha11mu |
?l;OOpaU'Ham'bt |
-qe'Hmpa 11IJI:JtCecmu Oa'HOpOa'H'btX |
me.!!, |
ozpa'HU"I,e'H'H'btX |
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n06epX'HOC11IJIMU: .
x2 y2 Z2
3.4.95.a2 + b2 = c2 ' Z = c.
3.4.96.x 2 + y2 = Z, X + y = a, x = 0, y = 0, z = 0.
3.4.97.HaitTM MOMeHT MHepu;MM qaCTM IIapa60JIOM,ll;a y2 + Z2 = 2cx, OTCe-
qeHHoit IIJIOCKOCThlO x = c, OTHOCMTeJIhHO OCM OX (Maccy IIPMHM-
MaTh, paBHoit e,ll;MHMu;e).
183
KOHTPOl1bHAH PA60TA
BapMaHT 1
1. IbMeHllTh IIOPH,ll;OK "HTerp"pOBaH"H
jdy 7r-iSin j(X, y) dx.
oarcsin y
2.Hail:Tll Maccy TpeyroJIhHllKa OAB, eCJIll 0(0,0), A(I, -1), B(I, 1), a IIJIOT-
HOCTh paBHa p(x,y) = y'x2 - y2.
3. |
Hail:Tll |
06'heM TeJIa, |
OrpaHll'IeHHoro |
IIJIOCKOCThlO |
Oxy, |
IJ;llJIllH,ll;POM |
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x 2 + y2 = |
4x II |
c<pep0il: x 2 + y2 + z2 |
= |
16 (BHYTpeHHero |
IIO |
OTHOIIIeHlllO |
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K |
IJ;llJIllH,ll;PY). |
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x: 'paCIIOJIO)KeHHoil: BHyTpll IJ;llJIllH,ll;pa |
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4. Hail:Tll IIJIOIu;a,ll;h IIOBepXHOCTll z= |
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x 2 + y2 = a2. |
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5. BhPIllCJIllTh TPOil:HOil: llHTerpaJI |
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!!!xdv, |
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= 0, y = lOx, |
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r,ll;e V - |
06JIaCTh, OrpaHll'IeHHM IIOBepXHOCTHMll x = 1, |
y |
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z = 0, z = xy. |
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BapMaHT 2 |
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1. BhPIllCJIllTh ,ll;BOil:HOil: llHTerpaJI |
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If |
~---- |
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1- ~ - |
-dxdy |
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y2 |
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eCJIll 06JIaCTh D OrpaHll'IeHa JIllHllHMll y = 0, |
y = ~v'a2 |
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x2 • |
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2. |
BhPIllCJIllTh llHTerpaJI |
!!r2 sin <p . r drd<p, |
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D |
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r,ll;e 06JIaCTh D OrpaHll'IeHa JIllHllHMll r = R, r = 2R sin <po |
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3. |
BhlqllCJIllTh |
Maccy |
IIJIOCKOil: |
<PllrYPhI, |
OrpaHllqeHHoil: |
JIeMHllcKaToil: |
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(x 2 + y2)2 = a2(x 2 _ y2), eCJIll ee IIJIOTHOCTh paBHa p(x, y) = xy'x2 + y2.
4. Hail:Tll MOMeHThI llHepIJ;llll O,ll;HOpO,ll;HOrO TpeyroJIhHllKa, OrpaHllqeHHoro IIpHMhIMll x + y = 2, 2x + y = 4, x = 0, OTHOCllTeJIhHO KOOp,ll;llHaTHhIX oceil:.
5. BhlqllCJIllTh TPOil:HOil: llHTerpaJI
!!!xy2eXYZ dxdydz, v
r,ll;e V - TeJIO, OrpaHllqeHHOe IIOBepXHOCTHMll x = 0, x = 2, y = 0, y = 3,
z = 1, z = 5.
184
Bapll1aHT 3
IIr3 drd<p,
D
eCJIH 06JIaCTh D OrpaHHqeHa JIeMHHcKaToit r2 = a2 cos 2<p H JIyqaMH <p = 0,
<p |
1r |
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= ~i" |
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2. IbMeHHTh nOpH,ll;OK HHTerpHpOBaHHH B nOBTopHOM HHTerpaJIe |
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a-Ja 2 _ y 2 |
a |
2a |
2a |
2a |
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I |
f(x,y) dx + I dy |
I |
f(x,y) dx + Idy |
I f(x,y) dx. |
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y2 |
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0 |
a+Ja2 -y2 |
0 |
y2 |
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2a |
3. HaitTH Maccy nJIacTHHhI D: (x - |
3)2 + y2 :::;; 1, x ~ 3, eCJIH ee nJIOTHOCTh B |
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TOqKe (x, y) paBHa Iyl. |
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4. |
BhIqHCJIHTh TPOitHOit HHTerpaJI |
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I I I y2 x cos xyz dxdydz,
V
eCJIH D - TeJIO, OrpaHHqeHHOe nOBepXHOCTHMH X = 1, x = 2, y = 1, y = 3,
z = 1, z = 4.
5. BhlqHCJIHTh Maccy TeJIa, OrpaHHqeHHOrO nJIOCKOCTHMH X = 0, y = 0, Z = 0,
X + y + z = 1, eCJIH nJIOTHOCTh B TOqKe (x, y, z) paBHa
1 |
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p(x,y,z) = (1 + X + y + Z)3 |
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Bapll1aHT 4 |
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IIr3 drd<p, |
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D |
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eCJIH D HMeeT BH,ll; °:::;; <p :::;; 21r, °:::;; r :::;; |
1 |
Vsin4 <p + cos4 <p
2. BhIqHCJIHTh nJIOIIIMh qaCTH nOBepXHOCTH C<pePhI x 2 + y2 + Z2 = 81, 3aKJUOqeHHYIO Me)KAY nJIOCKOCTHMH y = -5 H Y = 5.
3. HaitTH nJIOIIIMh <PHrYPhI, OrpaHHqeHHoit KPHBOit
(x + y - 3)2 + (2x - 3y + 5)2 = 49.
4. BhIqHCJIHTh TPOitHOit HHTerpaJI
III |
(1 |
dx |
4' |
V |
+ ~ + It + ~) |
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234 |
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185
eCJIH TeJIO V OrpaHHqeHO IIOBepXHOCTHMH x = 0, y = 0, Z = °H ~+ ~+ ~ = 1.
5. BblqHCJIHTb TpoiiHOii HHTerpaJI
!!!63(1 + 2..JY) dv, v
rAe TeJIO V OrpaHHqeHO IIOBepxHOCTHMH y = x, y = 0, x = 1, Z = 0, Z = xy.
...
rnaBa 4. KPLt1BOIlLt1HELt1HbIE
VI nOBEPXHOCTHblE Lt1HTErPAllbl
o
§ 1. KPVlBOJU1HEI7IHbll7l VlHTErPAll nEPBOrO POAA
OnpE!AeneH ....e Kp....Bon.... HeiiiHoro .... HTerpana nepBoro pOAa
=7 IIycTb B KalK.Il:Oit TOqKe rJIa.ll:Koit KPHBOit L = AB B rrJIOCKOCTH Oxy 3a.ll:aHa
HerrpepblBHaH <PYHKU;HH .Il:BYX rrepeMeHHblX f(x, y). IIpOH3BOJIbHO pa306beM KPHBYIO
L Ha n qacTeit TOqKaMH A = Mo, M I , M2, ... , Mn = B. 3aTeM Ha KalK.Il:Oit H3
rrOJIyqeHHbIX qaCTeit ~i BbI6epeM JII06yIO TOqKY Mi(Xi, jJ;) H COCTaBHM CYMMY
n
Sn = L f(Xi, jj;)flli,
i=l
r,!l;e flli = Mi-I Mi - .Il:JIHHa .Il:YrH ~i' IIoJIyqeHHaH CYMMa Ha3bIBaeTCH U'lt-
me2pa.!l!b'ltoti. CYMMOti. nepa020 poiJa .Il:JIH <PYHKU;HH f(x, y), 3MaHHoit Ha KPHBOit L.
0603HaqHM qepe3 d HaH6oJIbIIIYIO H3 .Il:JIHH .Il:Yr ~i (TaKHM 06Pa30M, d =
m~ flli). ECJIH rrpH d -+ 0 cYIII;ecTByeT rrpe.ll:eJI HHTerpaJIbHblX CYMM |
Sn |
(He |
• |
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3aBHcHIII;Hit OT crroco6a pa36HeHHH KpHBOit L Ha qaCTH H BbI60pa TOqeK |
Mi), |
TO |
2lTOT rrpe.ll:eJI Ha3bIBaeTCH "'puao.J!u'lteti.'It'btM U'ltme2pa.J!OM nepa020 poiJa OT <PYHKU;HH
f(x, y) rro KPHBOit L H 0603HaqaeTCH |
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jf(x, y) dl HJIH |
j f(x, y) dl. |
L |
AB |
MO:lKHO .Il:OKa3aTb, qTO eCJIH <PYHKU;HH f(x, y) HerrpepbIBHa, TO KPHBOJIHHeitHblit
HHTerpaJI
f!(x, y) dl
L
cym;ecTByeT. KPHBOJIHHeitHblit HHTerpaJI rrepBoro pO.ll:a 06JIMaeT CBoitcTBaMH, aHaJlOrHqHbIMH COOTBeTCTBYIOIII;HM CBoitcTBaM orrpe.ll:eJIeHHOrO HHTerpaJIa (aMHTHBHOCTb, JlHHeitHocTb, ou;eHKa MO,IJ,yJlH, TeopeMa 0 Cpe.ll:HeM). O.ll:HaKO eCTb OTJIHqHe:
j |
f(x, y) dl = j f(x, y) dl, |
AB |
BA |
T. e. KPHBOJIHHeitHblit HHTerpaJI rrepBoro pO.ll:a He 3aBHCHT OT HarrpaBJIeHHH HHTerpHpOBaHHH.
Bbl'-....l cneH ....e Kp....Bon....HeiiiHblx .... HTerpanoB nepBoro pOAa
BblqHCJIeHHe KPHBOJIHHeitHoro HHTerpaJIa rrepBoro pO.ll:a CBO.ll:HTCH K BblqHCJIeJUlIO orrpe.ll:eJIeHHoro HHTerpaJIa. A HMeHHO:
187
1. ECJIH KpHBaH L 3a,ll;aHa HerrpephIBHo ,ll;H<p<pepeHIJ;HpyeMoii <pYHKIJ;Heii y ::
=y(x), x E [a, b], TO
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b |
j |
f(x, y) dl = j f(x, y(x)) VI + (Y'(X))2 dx, |
L |
a |
rrpH 9TOM BhIpaJKeHHe dl = VI + (Y'(X))2 dx Ha3hIBaeTCH ,ll;H<p<pepeHIJ;HaJIOM .IJ..1lHHhI ,ll;yrH.
2. ECJIH KpHBaH L 3a,ll;aHa rrapaMeTpH'IeCKH,T. e. B BH,ll;e X = x(t), y = y(t), r,ll;e x(t), y(t) - HerrpephIBHO ,ll;H<p<pepeHIJ;HpyeMhIe <PYHKIJ;HH Ha HeKOTopOM OTpe3Ke
[a,,8], TO
f3
jf(x, y) dl = jf(x(t), y(t)) V(x' (t))2 + (y'(t))2 dt.
L Q
3TO paBeHCTBO pacrrpocTpaHHeTCH Ha cJIY'laiirrpOCTpaHCTBeHHOii KPHBOii L, 3a,ll;aHHOii rrapaMeTpH'IeCKH:X = X(t), Y = y(t), z = Z(t), t E [a, .B]. B 9TOM CJIY'Iae, eCJIH f(x, y, z) - HerrpephIBHaH <PYHKIJ;HH B,ll;OJIb KpHBOii L, TO
f3
jf(x, y, z) dl = jf[x(t), y(t), z(t)]v(x'(t))2 + (y'(t))2 + (Z'(t))2 dt.
L Q
3. ECJIH rrJIOCKaH KpHBaH L 3a,ll;aHa rrOJIHpHblM ypaBHeHHeM r = r (<p), <p E [a,.B],
TO
f3
jf(x, y) dl = jf(rcos<P,rsiII<P)Vr2 +r'2d<p.
L Q
npll1nO)l(eHII1R Kpll1BOnll1HeMHOrO II1HTerpana nepBOrO pOAa
1. ECJIH rrO,ll;bIHTerpaJIbHaH <PYHKIJ;HH paaHa e,ll;HHHIJ;e, TO KPHBOJIHHeiiHhIii HH-
TerpaJI
paaeH .IJ..1lHHe S KPHBOii L, T. e.
jdl = S.
L
2. TIyCTb B rrJIOCKOCTH Oxy 3a,ll;aHa rJIa,ll;KaH KpHBaH L, Ha KOTOpoii orrpe,ll;eJIeHa H HerrpephIBHa <PyHKIJ;HH ,ll;ByX rrepeMeHHblX z = f(x, y) ~ O. Tor,ll;a MO)KHO rrOCTpoHTb IJ;HJIHH,ll;pH'IeCKYIO rrOBepXHOCTb C HarrpaBJIHIOm;eii L H o6pa3YIOm;eii, rrapaJIJIeJIbHOii OCH Oz H 3aKJIIO'IeHHOiiMe)K,ll;y L H rrOBepXHOCTbIO z = f(x, y). TIJIOm;a,ll;b 9TOii IJ;HJIHH,ll;pH'IeCKOiirrOBepXHOCTH MO)KHO BhI'IHCJIHTbrro <popMYJIe
S= jf(x,y)dl.
L
188
3. ECJIH L = AB - M'aTepHaJIbHaHKpHBaH C nJIOTHOCTbIO, paBHoi!: p = p(x, y),
TO MaCCa 3TOi!: KPHBOi!: BbIQHCJ1HeTCH no <popMYJIe
m = j p(x,y)dl
AB
(ifju3U"I,eC1CuiJ, CM'b£CJt 1CpuaOJtUH.eiJ,H.01!O UH.me1!paJta nepa01!O poiJa).
4. CTaTHQeCKHe MOMeHTbl MaTepHaJIbHoi!: KPHBOi!: L OTHOCHTeJIbHO Koopp;HHaT-
HbIX ocei!: Ox H Oy COOTBeTCTBeHHO paBHbl
M", = j yp(x, y) dl, |
My = j xp(x, y) dl, |
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L |
L |
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me p(x, y) - nJIOTHOCTb pacnpep;eJIeHHH KPHBOi!: L, a Xc |
M", |
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m |
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Koopp;HHaTbl IIeHTpa TH:lKeCTH (IIeHTpa MacC) KPHBOi!: L. |
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5. MHTerpaJIbl |
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J", = j y2 p(x, y) dl, |
Jy = j x 2 p(x, y) dl, Jo= j(X2+ y2 )p(x,y)dl |
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L |
L |
L |
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Bblpa:lKaIOT MOMeHTbl HHepIIHH KPHBOi!: L C JIHHei!:Hoi!: nJIOTHOCTbIO p(x,y) OTHOCH-
TeJIbHO ocei!: Ox, Oy H HaQaJIa Koopp;HHaT COOTBeTCTBeHHO.
4.1.1. BbPUICJUITb KPHBOJIMHeiiHblii MHTerpa.rr
jW dl ,
L
rp;e L - .rr.yra rrapa60JIbI y2 = 2x, 3aKJIlOQeHHaH Me)K.rr.y TOQKaMH
(2,2) H (8,4).
a Haii,IIeM ,IIH<p<pepeHIIHa.rr .rr.ym dl ,IIJIH KpMBOii Y = ffx. MMeeM
y' = vk'dl = VI + (y')2 dx = VI + 2~dx.
CJIep;OBaTeJIbHO, p;aHHblii HHTerpa.rr paBeH
~dl= f~Vl+ 1 dx = j xv'I+2X dx = |
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jy |
ffx |
2x |
2x |
L |
2 |
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8 |
8 |
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= ~ j vI + 2xdx = ~ ·l(1 + 2X)3/212 = i(17vTI - 5v'5). • |
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2 |
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4.1.2. |
BbIQMCJIMTb KPMBOJIHHeiiHblii HHTerpa.rr |
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j(x2 + y3) dl, |
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L |
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r,IIe L - |
KOHTYP |
TpeyroJIbHHKa ABO C BepIIIMHaMM A(I,O), |
B(O, 1), 0(0,0) (pMC. 43).
189